\(\int (a+\frac {b}{x}) \sqrt {x} \, dx\) [1649]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \left (a+\frac {b}{x}\right ) \sqrt {x} \, dx=2 b \sqrt {x}+\frac {2}{3} a x^{3/2} \]

[Out]

2/3*a*x^(3/2)+2*b*x^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int \left (a+\frac {b}{x}\right ) \sqrt {x} \, dx=\frac {2}{3} a x^{3/2}+2 b \sqrt {x} \]

[In]

Int[(a + b/x)*Sqrt[x],x]

[Out]

2*b*Sqrt[x] + (2*a*x^(3/2))/3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b}{\sqrt {x}}+a \sqrt {x}\right ) \, dx \\ & = 2 b \sqrt {x}+\frac {2}{3} a x^{3/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \left (a+\frac {b}{x}\right ) \sqrt {x} \, dx=\frac {2}{3} \sqrt {x} (3 b+a x) \]

[In]

Integrate[(a + b/x)*Sqrt[x],x]

[Out]

(2*Sqrt[x]*(3*b + a*x))/3

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68

method	result	size
gosper	\(\frac {2 \left (a x +3 b \right ) \sqrt {x}}{3}\)	\(13\)
trager	\(\left (\frac {2 a x}{3}+2 b \right ) \sqrt {x}\)	\(13\)
risch	\(\frac {2 \left (a x +3 b \right ) \sqrt {x}}{3}\)	\(13\)
derivativedivides	\(\frac {2 a \,x^{\frac {3}{2}}}{3}+2 b \sqrt {x}\)	\(14\)
default	\(\frac {2 a \,x^{\frac {3}{2}}}{3}+2 b \sqrt {x}\)	\(14\)

[In]

int((a+b/x)*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(a*x+3*b)*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \left (a+\frac {b}{x}\right ) \sqrt {x} \, dx=\frac {2}{3} \, {\left (a x + 3 \, b\right )} \sqrt {x} \]

[In]

integrate((a+b/x)*x^(1/2),x, algorithm="fricas")

[Out]

2/3*(a*x + 3*b)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{x}\right ) \sqrt {x} \, dx=\frac {2 a x^{\frac {3}{2}}}{3} + 2 b \sqrt {x} \]

[In]

integrate((a+b/x)*x**(1/2),x)

[Out]

2*a*x**(3/2)/3 + 2*b*sqrt(x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+\frac {b}{x}\right ) \sqrt {x} \, dx=\frac {2}{3} \, {\left (a + \frac {3 \, b}{x}\right )} x^{\frac {3}{2}} \]

[In]

integrate((a+b/x)*x^(1/2),x, algorithm="maxima")

[Out]

2/3*(a + 3*b/x)*x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+\frac {b}{x}\right ) \sqrt {x} \, dx=\frac {2}{3} \, a x^{\frac {3}{2}} + 2 \, b \sqrt {x} \]

[In]

integrate((a+b/x)*x^(1/2),x, algorithm="giac")

[Out]

2/3*a*x^(3/2) + 2*b*sqrt(x)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.63 \[ \int \left (a+\frac {b}{x}\right ) \sqrt {x} \, dx=\frac {2\,\sqrt {x}\,\left (3\,b+a\,x\right )}{3} \]

[In]

int(x^(1/2)*(a + b/x),x)

[Out]

(2*x^(1/2)*(3*b + a*x))/3